WASHINGTON (CNN) -- Vice President Dick Cheney should testify before Congress about his role in the leaking of a CIA agent's identity, former White House spokesman Scott McClellan told members of the House Judiciary Committee on Friday.

"The vice president has information that has not been shared publicly," McClellan said in response to a question from Rep. Debbie Wasserman Schultz, a Florida Democrat, about whom Congress should question in connection with the leaking of Valerie Plame Wilson's name to the media.

"You could go down the list: Karl Rove, Ari Fleischer ..." McClellan added, referring to President Bush's former political adviser and first White House spokesman.

McClellan said he does not think the president knew in advance about the Plame Wilson leak.

Earlier, he told the committee that suspicion still surrounds Cheney's office because of unanswered questions about the leak.

McClellan also also said he could not rule out that Cheney had ordered his former chief of staff, Lewis "Scooter" Libby, to leak Plame Wilson's identity. A later investigation showed that Libby, in fact, had been involved in the leaking of the CIA operative's name.

The FBI and a federal grand jury later investigated the leaking of Plame Wilson's identity. No one was convicted for the actual leak, but Libby was later convicted of perjury, lying to federal investigators and obstruction of justice in 2007.

Don't Miss

"I think that [special prosecutor] Patrick Fitzgerald stated it well when he talked about the cloud that was remaining over the vice president's office because of Scooter Libby's actions that led to his conviction on four counts, I guess," McClellan said Friday.

"But there's a lot of suspicion there because there are questions that have never been answered, despite the fact that we said at some point we would address these issues," he said.

"I do not think the president had any knowledge" of the revelation of Plame Wilson's identity, McClellan said. "In terms of the vice president, I do not know."

McClellan also testified that he was asked at the behest of Bush and Cheney to provide assurances to the media that Libby was not involved in the leak of Plame Wilson's identity.

"I was reluctant to do it, but ... I talked with [former White House chief of staff] Andy Card," McClellan said. He also talked to Libby.

Libby "assured me in unequivocal terms that he was not, meaning the leaking of Valerie Plame's identity to any reporters, and then I contacted reporters to let them know about that information," McClellan said. "But it was Andy Card that had directed me to do that, at the request of the president and vice president." Watch McClellan detail his talks with Libby »

"This matter continues to be investigated by Congress because of what the White House has chosen to conceal from the public," McClellan said. "Despite assurances that the administration would discuss the matter once the special counsel had completed his work, the White House has sought to avoid public scrutiny and accountability.

"I do not know whether a crime was committed by any of the administration officials who revealed Valerie Plame's identity to reporters," he said. "Nor do I know if there was an attempt by any person or persons to engage in a cover-up during the investigation. I do know that it was wrong to reveal her identity, because it compromised the effectiveness of a covert official for political reasons."

Two senior administration officials say the White House decided not to invoke executive privilege to stop McClellan's testimony because "there's nothing new."

Members of the House Judiciary Committee believe that McClellan, who was the president's press secretary before leaving the White House in 2006, would be able to provide valuable insight into the leaking of Plame Wilson's name as well as a number of issues that the committee is investigating.

In his book, "What Happened: Inside the Bush White House and Washington's Culture of Deception," McClellan said he was specifically lied to about White House staff members' involvement in the leaking of Plame Wilson's identity -- including Rove and Libby -- in 2003. Read excerpts of McClellan's book »

"When I was knowingly misled but only learned that much later, that's really when I started to become disillusioned at the White House."

Libby was convicted of lying to investigators and a grand jury about his involvement in the CIA leak case. Bush commuted his sentence.

"I'm glad to share my views," McClellan said. "I think I've made them very clear in the book. ... Essentially, everything I know on that leak episode is written in the book -- what I was told by Karl Rove and Scooter Libby."

The committee is also looking into the use of prewar intelligence and whether politics was behind the firing of eight U.S. attorneys in 2006.

As White House spokesman, McClellan defended Bush's policies during much of the war in Iraq, the aftermath of Hurricane Katrina and the scandal that followed the leak of Plame Wilson's identity.

He said that the latter episode was a "defining moment that caused me to become dismayed and disillusioned with the way things were going in Washington, D.C."

McClellan's critics say he was not in a position to know details of the innermost workings of the White House, particularly as he was deputy press secretary for domestic affairs in the time before the Iraq war.

In his memoir, McClellan says the administration became mired in "propaganda" and political spin and played loose with the truth at times.